The 2-Hilbert Space of a Prequantum Bundle Gerbe

TL;DR

This paper constructs a 2-Hilbert space for torsion line bundle gerbes, extending prequantisation concepts to higher categorical structures and exploring their properties and relations to classical prequantisation.

Contribution

It introduces a novel 2-Hilbert space framework for line bundle gerbes, including a dual functor and compatibility with transgression, advancing higher prequantisation theory.

Findings

The 2-Hilbert space is semisimple and abelian.

The dual functor induces a closed structure on morphisms.

Compatibility of the transgression functor with higher structures is demonstrated.

Abstract

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf's version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantisation. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant-Souriau prequantisation in this setting, including its dimensional reduction to ordinary prequantisation.